Work

An Efficient Temporal Two-Mesh Compact ADI Method for Nonlinear Schrödinger Equations with Error Analysis

Year: 2025

Type: article

Abstract: In this article, we present an efficient numerical strategy for the two-dimensional nonlinear Schrödinger equation, focusing on its development and analysis. Our approach begins with proposing a nonlinear, energy-conservative, fourth-order, compact, alternating-direction, implicit (ADI) scheme. To boost efficiency when solving the associated nonlinear system, we then implement this scheme using a temporal two-mesh (TTM) algorithm. Under discretization with coarse time step τC, fine time step τF, and spatial mesh size h, the numerical scheme exhibits a convergence rate of order O(τC4+τF2+h4) in both the discrete L2-norm and H1-norm. To facilitate the convergence analysis under fine time discretization, we propose a novel technique along with several supporting lemmas that enable the estimation of the discrete L4-norm error term over the temporal coarse mesh. Numerical experiments are then performed to validate the theoretical results and demonstrate the effectiveness of the proposed algorithm. The numerical results show that the new algorithm produces highly accurate results and preserves the conservation laws of mass and energy. Compared with the fully nonlinear compact ADI scheme, it reduces computational time while maintaining accuracy.

Source: Axioms

DOI: 10.3390/axioms14110777

Authors: Siriguleng He, Eerdun Buhe, Chelimuge Bai

Institution: Hohhot Minzu College

Language: English

Topic: Numerical methods for differential equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funder: National Natural Science Foundation of China